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arxiv: 1906.10023 · v1 · pith:XJZC7N4Znew · submitted 2019-06-24 · 🪐 quant-ph · math-ph· math.MP

A Class of PPT Entangled States Arbitrary Far From Separable States

Adam Rutkowski , Micha{\l} Studzi\'nski This is my paper

Pith reviewed 2026-05-25 17:20 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords PPT entangled statesmultipartite entanglementtrace distanceseparable statespositive partial transpositionquantum statesentanglement witnesses
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The pith

An explicit family of multipartite PPT entangled states can be made arbitrarily far from separable states in trace distance, with the distance growing as local dimension increases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a specific class of multipartite quantum states that remain entangled while satisfying the positive partial transposition criterion across every possible cut. Within this class the authors isolate a subfamily whose trace distance to the nearest separable state grows without bound as the dimension of each local Hilbert space is increased. The construction achieves this separation directly on a single copy of the state rather than by tensoring multiple copies of a lower-dimensional example. A reader would care because it shows that the geometric gap between PPT entanglement and separability need not stay bounded in multipartite systems.

Core claim

We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction we do not have to use many copies of initial state living on the smaller space to boost the trace distance as in the previous attempts to this problem.

What carries the argument

An explicit multipartite PPT entangled state family whose trace-distance lower bound to separability holds for arbitrary local dimension without parameter retuning.

If this is right

  • The trace distance lower bound holds uniformly for the subfamily at every local dimension.
  • The distance to separability increases with local Hilbert-space dimension in the multipartite setting.
  • The separation is achieved without tensoring multiple copies of a lower-dimensional state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction may yield states whose distance to separability remains large under small perturbations of the local dimension.
  • One could test whether the growth rate of the distance is linear or faster in the local dimension by evaluating the bound at successive integer values of d.
  • The family supplies concrete examples that could be used to probe whether other entanglement measures, such as robustness or negativity, also diverge with dimension.

Load-bearing premise

The constructed states remain PPT and entangled for every local dimension while the trace-distance lower bound holds without additional constraints.

What would settle it

An explicit calculation for some local dimension d showing that every state in the family has trace distance to the separable set bounded above by a fixed constant independent of d.

read the original abstract

In this paper we show an explicit construction of multipartite class of entangled states with the PPT (Positive Partial Transposition) property in every cut. We investigate properties of this class of states focusing on the trace distance from the set of separable states. We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue, that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction is we do not have to use many copies of initial state living on the smaller space to boost the trace distance as in the previous attempts to this problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents an explicit construction of a multipartite family of PPT entangled states (positive under partial transposition across every cut) and isolates a subclass for which the trace distance to the set of separable states is bounded from below by a quantity that grows with the local Hilbert-space dimension d. The construction is claimed to achieve this separation without tensoring multiple copies of a lower-dimensional seed state.

Significance. If the explicit family and the accompanying lower-bound proof are correct, the work supplies a concrete, parameter-uniform example showing that multipartite PPT entanglement can be made arbitrarily distant from separability simply by increasing local dimension. This is a useful addition to the literature on the geometry of the PPT and separable sets in the multipartite regime.

major comments (2)
  1. [Construction and distance-bound sections] The central claim requires an explicit definition of the state family ρ_d together with a uniform proof that (i) ρ_d remains PPT across every bipartition and (ii) the trace-distance lower bound grows with d, all without any d-dependent free parameters that could be adjusted post hoc. The manuscript must exhibit the concrete matrices or projectors and the calculation establishing the bound for arbitrary d.
  2. [Proof of trace-distance lower bound] The argument that the distance increases with dimension must be shown to survive for every d without retuning; if the proof introduces any auxiliary parameters whose values are chosen differently for different d, the claimed uniformity fails.
minor comments (1)
  1. [Abstract] Abstract contains a grammatical error: 'In our construction is we do not have to use' should be rephrased.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the significance of our results and for the detailed comments on the explicitness and uniformity of the construction. We address each major comment below and indicate the revisions we will make to improve clarity without altering the core claims.

read point-by-point responses

  1. Referee: [Construction and distance-bound sections] The central claim requires an explicit definition of the state family ρ_d together with a uniform proof that (i) ρ_d remains PPT across every bipartition and (ii) the trace-distance lower bound grows with d, all without any d-dependent free parameters that could be adjusted post hoc. The manuscript must exhibit the concrete matrices or projectors and the calculation establishing the bound for arbitrary d.

    Authors: The manuscript presents an explicit construction of the family ρ_d in the main text, given as a fixed convex combination of product and entangled projectors on the multipartite space that is defined uniformly for any local dimension d. The PPT property across all bipartitions is established by direct verification that each partial transpose is positive semidefinite, with the algebraic steps independent of d. The trace-distance lower bound is obtained via a fixed entanglement witness whose expectation value produces a quantity that grows with d; the same witness and bounding technique apply for every d without auxiliary parameters. To address the request for greater concreteness, we will add an appendix displaying the explicit matrix form of ρ_d and the key steps of the bound calculation for arbitrary d. revision: partial

  2. Referee: [Proof of trace-distance lower bound] The argument that the distance increases with dimension must be shown to survive for every d without retuning; if the proof introduces any auxiliary parameters whose values are chosen differently for different d, the claimed uniformity fails.

    Authors: The lower-bound argument is formulated without any d-dependent auxiliary parameters or retuning. The witness operator is chosen once, independently of d, and the subsequent trace-norm estimate relies only on the dimension-dependent support of the state itself; the identical sequence of inequalities holds for all d ≥ 2. We will add a short paragraph in the revised version explicitly stating that no parameter retuning occurs and that the proof is verbatim for every d. revision: partial

Circularity Check

0 steps flagged

Explicit construction supplies independent lower bound

full rationale

The manuscript supplies an explicit family of states whose PPT property across cuts and trace-distance lower bound are derived directly from the given matrices/projectors. No parameter is fitted to the target distance, no self-citation chain is invoked to force uniqueness, and the increase with local dimension follows from the closed-form expression rather than from re-labeling or re-use of the same quantity. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit state parametrization, so free parameters, axioms, and invented entities cannot be enumerated; the construction is presumed to rest on standard definitions of PPT, entanglement, and trace distance.

pith-pipeline@v0.9.0 · 5637 in / 1023 out tokens · 26899 ms · 2026-05-25T17:20:14.897831+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bound entanglement in symmetric random induced states

    quant-ph 2025-10 unverdicted novelty 6.0

    Symmetric random induced states yield PPT bound entanglement with probability close to 1 for N>3 qubits via two partial tracing constructions.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

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